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A geometric formulation of GENERIC stochastic differential equations

Mark A. Peletier, Marcello Seri

TL;DR

This work addresses the lack of coordinate invariance in stochastic GENERIC formulations by introducing a coordinate-free geometric framework for GENERIC SDEs on manifolds. It defines a gGENERIC SDE using a degenerate Poisson structure ${\mathsf J}$, a degenerate co-metric ${\mathsf K}$, a unimodular volume form ${\nu}$, and energy/entropy functionals $E,S$, yielding a generator $L f = {\mathsf J}(dE, df) + {\mathsf K}(dS, df) + \Delta_{\mathcal H} f$ and a Stratonovich SDE driven by horizontal vector fields; the model preserves the Boltzmann-type measure $e^{S}\nu$ and exactly conserves energy almost surely. It shows a consistent reduction to deterministic GENERIC in the zero-noise limit and casts the Fokker-Planck equation as a deterministic GENERIC evolution on an infinite-dimensional manifold, linking stochastic dynamics to coarse-graining and potential quantum extensions. The framework cleanly separates ambient geometry from system-specific data, enabling robust analytic and numerical methods and providing a principled basis for future extensions to coarse-grained and quantum systems.

Abstract

We propose a coordinate-invariant geometric formulation of the GENERIC stochastic differential equation, unifying reversible Hamiltonian and irreversible dissipative dynamics within a differential-geometric framework. Our construction builds on the classical GENERIC or metriplectic formalism, extending it to manifolds by introducing a degenerate Poisson structure, a degenerate co-metric, and a volume form satisfying a unimodularity condition. The resulting equation preserves a particular Boltzmann-type measure, ensures almost-sure conservation of energy, and reduces to the deterministic GENERIC/metriplectic formulation in the zero-noise limit. This geometrization separates system-specific quantities from the ambient space, clarifies the roles of the underlying structures, and provides a foundation for analytic and numerical methods, as well as future extensions to quantum and coarse-grained systems.

A geometric formulation of GENERIC stochastic differential equations

TL;DR

This work addresses the lack of coordinate invariance in stochastic GENERIC formulations by introducing a coordinate-free geometric framework for GENERIC SDEs on manifolds. It defines a gGENERIC SDE using a degenerate Poisson structure , a degenerate co-metric , a unimodular volume form , and energy/entropy functionals , yielding a generator and a Stratonovich SDE driven by horizontal vector fields; the model preserves the Boltzmann-type measure and exactly conserves energy almost surely. It shows a consistent reduction to deterministic GENERIC in the zero-noise limit and casts the Fokker-Planck equation as a deterministic GENERIC evolution on an infinite-dimensional manifold, linking stochastic dynamics to coarse-graining and potential quantum extensions. The framework cleanly separates ambient geometry from system-specific data, enabling robust analytic and numerical methods and providing a principled basis for future extensions to coarse-grained and quantum systems.

Abstract

We propose a coordinate-invariant geometric formulation of the GENERIC stochastic differential equation, unifying reversible Hamiltonian and irreversible dissipative dynamics within a differential-geometric framework. Our construction builds on the classical GENERIC or metriplectic formalism, extending it to manifolds by introducing a degenerate Poisson structure, a degenerate co-metric, and a volume form satisfying a unimodularity condition. The resulting equation preserves a particular Boltzmann-type measure, ensures almost-sure conservation of energy, and reduces to the deterministic GENERIC/metriplectic formulation in the zero-noise limit. This geometrization separates system-specific quantities from the ambient space, clarifies the roles of the underlying structures, and provides a foundation for analytic and numerical methods, as well as future extensions to quantum and coarse-grained systems.

Paper Structure

This paper contains 22 sections, 12 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. GENERIC
  3. GENERIC SDEs
  4. Stochastic differential equations on manifolds
  5. Definition of an SDE through Stratonovich integration
  6. Generator characterization
  7. Geometrization
  8. Components of the geometric GENERIC SDE
  9. Sub-Riemannian Brownian motion generated by ${\mathsf K}$
  10. Defining the geometric GENERIC SDE
  11. Properties of the process
  12. The Euclidean case
  13. Reduction to the Euclidean case
  14. Coordinate transformations in the Euclidean GENERIC SDE
  15. The Fokker-Planck equation
  16. ...and 7 more sections

Key Result

Lemma 3.1

Assume that $E,S,J,$ and $K$ satisfy conditions cond:generic-metric--cond:generic-degeneracy of the deterministic GENERIC equation in Section sec:generic and that eq:fluctuation-dissipation holds. Then the evolution deterministically preserves the energy $E$, that is, for any $X_0\in{\mathbb R}^d$,

Theorems & Definitions (36)

  • Remark 2.1: Notational convention for derivatives in ${\mathbb R}^d$
  • Remark 2.2: Historical remarks and terminology
  • Remark 2.3: Finite dimensions
  • Remark 2.4: Non-quadratic dissipation
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.3
  • Definition 4.1: Manifold SDE; Thalmaier16
  • Definition 4.2: Solution of the SDE; Thalmaier16
  • Theorem 4.3: Existence and uniqueness of solutions of the SDE; Thalmaier16
  • ...and 26 more